The advantage of small machines in a stochastic fluid production process

We consider a stochastic fluid production model, where m machines which are subject to breakdown and repair, produce a fluid at ratep > 0 per machine if it is working. This fluid is fed into an infinite buffer with stochastic output rate. Under the assumption that the machine processes are independent and identically distributed, we prove that the buffer content at timet is less or equal in the increasing convex ordering to the buffer content at time t of a model withm m machines and production ratep =m/m p. This formulation includes a conjecture posed by Mitra [6]. More-over, it is shown how to extend this result to Brownian flow systems, systems obtained by diffusion approximation and simple stochastic flow networks like tandem buffer and assembly systems.     

Monotonicity and bounds for convex stochastic control models

We consider a general convex stochastic control model. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the transition kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on convex stochastic orderings of probability measures. We derive several interesting sufficient conditions of these ordering concepts, where we make also use of the Blackwell ordering. The structural results are illustrated by partially observed control models and Bayesian information models.     

Identification of ground water flow patterns using particle models

Particle models are often used to simulate transport processes in ground water. The ground water flow pattern is one of the driving parameters of the transport model. In this paper a parameter identification algorithm is developed for particle type models to identify the underlying flow pattern from concentration measurements. The estimation problem is solved with a gradient based algorithm. For each generated particle track, the adjoint track is determined to efficiently compute gradient of the criterion.     

Optimal order for two servers in tandem

We consider two servers (serveri, i=1, 2) in tandem for which the order of servers can be changed. Server 1 has a general service time distribution and server 2 has either its shifted or truncated distribution. This permits that the service times at the two servers are overlapping. An unlimited queue is allowed in front of the first server. For the systems having zero buffer capacity between the servers, we show that the sojourn time of every customer is stochastically minimized under any arrival process if server 2 is first. For the systems with infinite buffer capacity and a Poisson arrivals, we show that this order of servers minimizes mean customer delay when traffic is light. Several numerical examples are presented to demonstrate that this optimal order is invariant under any arrival process (the interarrival times are i.i.d. r.v.'s) and mild traffic condition.Research funded by NEC Corporation C & C Laboratory.     

Parallel simulation of individual-based, physiologically structured population models

A general scheme for parallel simulation of individual-based, structured population models is proposed. Algorithms are developed to simulate such models in a parallel computing environment. The simulation model consists of an individual model and a population model that incorporates the individual dynamics. The individual model is a continuous time representation of organism life history for growth with discrete allocations for reproductive processes. The population model is a continuous time simulation of a nonlinear partial differential equation of extended McKendrick-von Foerster-type.

As a prototypical example, we show that a specific individual-based, physiologically structured model for Daphnia populations is well suited for parallelization, and significant speed-ups can be obtained by using efficient algorithms developed along our general scheme. Because the parallel algorithms are applicable to generic structured populations which are the foundation for populations in a more complex community or food-web model, parallel computation appears to be a valuable tool for ecological modeling and simulation.     

Coupling strategies for the numerical simulation of blood flow in deformable arteries by 3D and 1D models

The fluid structure interaction mechanism in vascular dynamics can be described by either 3D or 1D models, depending on the level of detail of the flow and pressure patterns needed for analysis. A successful strategy that has been proposed in the past years is the so-called geometrical multiscale approach, which consists of coupling both 3D and 1D models so as to use the former only in those regions where details of the fluid flow are needed and describe the remaining part of the vascular network by the simplified 1D model.In this paper we review recently proposed strategies to couple the 3D and 1D models, and within the 3D model, to couple the fluid and structure sub-problems. The 3D/1D coupling strategy relies on the imposition of the continuity of flow rate and total normal stress at the interface. On the other hand, the fluid–structure coupling strategy employs Robin transmission conditions. We present some numerical results and show the effectiveness of the new approaches.     

Numerical simulation of generalized newtonian blood flow past a couple of irregular arterial stenoses

This paper looked at the numerical investigations of the generalized Newtonian blood flow through a couple of irregular arterial stenoses. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mild stenosed artery, so that the flow effectively becomes two‐dimensional. The Marker and Cell (MAC) method is developed for the governing unsteady generalized Newtonian equations in staggered grid for viscous incompressible flow in the cylindrical polar co‐ordinates system. The derived pressure‐Poisson equation was solved using Successive‐Over‐Relaxation (S.O.R.) method and the pressure‐velocity correction formulae have been derived. Computations are performed for the pressure drop, the wall shear stress distribution and the separation region. The presented computations show that in comparison to the corresponding Newtonian model the generalized Newtonian fluid experiences higher pressure drop, lower peak wall shear stress and smaller separation region. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 960–981, 2011     

Buffer Overflow Probabilities for a Markov-Modulated Fluid Model

Markov modulated fluid models are widely used in modelling communications and computer systems. In the AMS (Annick, Mitra, Sohndi) model, heterogeneous, bursty sources modeled by multidimensional Markov processes are superimposed or multiplexed together to drive a fluid buffer. The performance of the system is measured by the steady state probability that the buffer exceeds a high level. The exact solution to this problem derived by AMS requires too much computation to be used on-line. Here we derive an upper bound for the above probability which is fast to compute and accurate enough for practical use.     

Analysis of traffic flow by the finite element method

A finite elernent methodology is developed for the numerical solution of traffic flow problems encountered in arterial streets. The simple continuum traffic flow model consisting of the equation of continuity and an equilibrium flow-density relationship is adopted. A Galerkin type finite element method is used to formulate the problem in discrete form and the solution is obtained by a step-by-step time integration in conjunction with the Newton-Raphson method. The proposed finite element methodology, which is of the shock capturing type, is applied to flow traffic problems. Two numerical examples illustrate the method and demonstrate its advantages over other analytical or numerical techniques.     

Products of Beta matrices and sticky flows

A discrete model of Brownian sticky flows on the unit circle is described: it is constructed with products of Beta matrices on the discrete torus. Sticky flows are defined by their moments which are consistent systems of transition kernels on the unit circle. Similarly, the moments of the discrete model form a consistent system of transition matrices on the discrete torus. A convergence of Beta matrices to sticky kernels is shown at the level of the moments. As the generators of the n-point processes are defined in terms of Dirichlet forms, the proof is performed at the level of the Dirichlet forms. The evolution of a probability measure by the flow of Beta matrices is described by a measure-valued Markov process. A convergence result of its finite dimensional distributions is deduced.Mathematics Subject Classification (2000):60J27, 60J35, 60G09     

Sub-stochastic matrix analysis for bounds computation—Theoretical results

Performance evaluation of complex systems is a critical issue and bounds computation provides confidence about service quality, reliability, etc. of such systems. The stochastic ordering theory has generated a lot of works on bounds computation. Maximal lower and minimal upper bounds of a Markov chain by a st-monotone one exist and can be efficiently computed. In the present work, we extend simultaneously this last result in two directions. On the one hand, we handle the case of a maximal monotone lower bound of a family of Markov chains where the coefficients are given by numerical intervals. On the other hand, these chains are sub-chains associated to sub-stochastic matrices. We prove the existence of this maximal bound and we provide polynomial time algorithms to compute it both for discrete and continuous Markov chains. Moreover, it appears that the bounding sub-chain of a family of strictly sub-stochastic ones is not necessarily strictly sub-stochastic. We establish a characterization of the families of sub-chains for which these bounds are strictly sub-stochastic. Finally, we show how to apply these results to a classical model of repairable system. A forthcoming paper will present detailed numerical results and comparison with other methods.     

On the distribution of the number of packets in the fluid flow approximation of packet arrival streams

Fluid flow approximations are widely used for approximating models of communication systems where packet arrival streams are generated in a regular manner over certain intervals (constant rate). The appropriate mathematical model for describing those bursty arrival streams in the fluid flow framework are the well-known Markov modulated rate processes (MMRP). The paper deals with the distribution of the numberN(t) of packets in the interval [0,t] of MMRP. For two-state MMRPs and their superpositions we derive formulas for the distribution ofN(t) and its density. Further we give asymptotic results. The presented numerical results and simulation studies illustrate the goodness of the fluid flow approximation and show that the proposed numerical algorithms work well even in the case of multiplexing a large number of burst silence sources.This work was partially supported by a grant from the Deutsche Bundespost TELEKOM.     

Modelling and analysis of Markovian continuous flow systems with a finite buffer

In this study, a Markovian fluid flow system with two stages separated by a finite buffer is considered. Fluid flow models have been analyzed extensively to evaluate the performance of production, computer, and telecommunication systems. Recently, we developed a methodology to analyze general Markovian continuous flow systems with a finite buffer. The flexibility of this methodology allows us to analyze a wide range of systems by specifying the transition rates and the flow rates associated with each state of each stage. In this study, in order to demonstrate the applicability of our methodology, we model and analyze a range of models studied in the literature. The examples we analyze as special cases of our general model include systems with phase-type failure and repair-time distributions, systems with machines that have multiple up and down states, and systems with multiple unreliable machines in series or parallel in each stage. For each case, the Markovian model is developed, the transition and flow rates are determined, and representative numerical results are obtained by using our methodology.     

Fast deterministic approximation for the multicommodity flow problem

In this paper we consider an optimization version of the multicommodity flow problem which is known as the maximum concurrent flow problem. We show that an approximate solution to this problem can be computed deterministically using O(k(ε ⁻² + logk) logn) 1-commodity minimum-cost flow computations, wherek is the number of commodities,n is the number of nodes, andε is the desired precision. We obtain this bound by proving that in the randomized algorithm developed by Leighton et al. (1995) the random selection of commodities can be replaced by the deterministic round-robin without increasing the total running time. Our bound significantly improves the previously known deterministic upper bounds and matches the best known randomized upper bound for the approximation concurrent flow problem. A preliminary version of this paper appeared inProceedings of the 6th ACM-SIAM Symposium on Discrete Algorithms, San Francisco CA, 1995, pp. 486–492.     

A class of renewal Interrupted Poisson Processes and applications to queueing systems

Switched Poisson Processes and Interrupted Poisson Processes are often employed to characterize traffic streams in distributed computer and communications systems, especially in investigations of overflow processes in telecommunication networks. With these processes, input streams having inter-segment correlations and high variance as well as state-dependent traffic can properly be modelled. In this paper we first derive an approximation method to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption. As a special case of this class of processes, the class of Interrupted Poisson processes is also included in the investigation. As a result, a generalization of the well-known class of Interrupted Poisson processes is obtained. It is shown that the renewal property is also given for this general class of Interrupted Poisson processes having generally distributed off-phase. To illustrate the accuracy of the presented renewal approximation of Generalized Switched Poisson processes and to show the major properties of the General Interrupted Poisson processes, applications to some basic queueing systems are discussed by means of numerical results.This work was done while the author was with Institute of Communications Switching and Data Technics, University of Stuttgart, Seidenstrasse 36, D-7000 Stuttgart 1, FRG.     

单自由度摩擦系统离散模型

发展了两种随机离散数学模型:导出了一个以二维平均映射描述的随机模型,并建立了一个概率预报模型.通过实例对不同模型进行了比较,对于平均映射模型,分岔图指出了外噪声对系统性质的影响,通过符号动力学方法分析指出概率预报模型的随机性质.     

Object-oriented modelling and simulation of heat exchangers with finite element methods

The paper describes the derivation of finite-element models of one-dimensional fluid flows with heat transfer in pipes, using the Galerkin/least-squares approach. The models are first derived for one-phase flows, and then extended to homogeneous two-phase flows. The resulting equations have then been embedded in the context of object-oriented system modelling; this allows one to combine the fluid flow model with a model for other phenomena such as heat transfer, as well as with models of other discrete components such as pumps or valves, to obtain complex models of heat exchangers. The models are then validated by simulating a typical heat exchanger plant.     

Feedback flow control employing local dynamical modelling with wavelets

In this paper, we utilize wavelet transform to obtain dynamical models describing the behaviour of fluid flow in a local spatial region of interest. First, snapshots of the flow are obtained from experiments or from computational fluid dynamics (CFD) simulations of the governing equations. A wavelet family and decomposition level is selected by assessing the reconstruction success under the resulting inverse transform. The flow is then expanded onto a set of basis vectors that are constructed from the wavelet function. The wavelet coefficients associated with the basis vectors capture the time variation of the flow within the spatial region covered by the support of the basis vectors. A dynamical model is established for these coefficients by using subspace identification methods. The approach developed is applied to a sample flow configuration on a square domain where the input affects the system through the boundary conditions. It is observed that there is good agreement between CFD simulation results and the predictions of the dynamical model. A controller is designed based on the dynamical model and is seen to be successful in regulating the velocity of a given point within the region of interest.